Coverings, densities, and Vitali theorems

The five-radius covering lemma selects a disjoint subfamily from a family of balls with uniformly bounded radii in Euclidean space, so the union of the original balls is covered by fivefold dilations of selected ones. Countability follows from separability for the disjoint positive-radius balls. The enlargement factor makes this a covering estimate; it does not say that the selected original balls already cover almost every point. Such estimates are central in maximal-function and density arguments.

The Vitali covering theorem gives a different conclusion. For a finite-measure Lebesgue-measurable set and a fine family of balls centered at its points, with arbitrarily small available radii at every point, one can select disjoint balls covering the set up to a Lebesgue-null remainder. The fine-cover hypothesis and the declared measure matter. One must not replace them by a single fixed-radius cover or assume that a Euclidean Lebesgue statement automatically holds for every metric measure space.

If f is locally Lebesgue integrable in Euclidean n-space, the average absolute difference from f at a point tends to zero at almost every point. Consequently, for the absolutely continuous measure with density f, the ball-mass ratio tends to f almost everywhere. The denominator is the volume of an n-ball. Both local integrability and the reference Lebesgue measure enter this statement; it is not a claim that every chosen point is a Lebesgue point.

The value assigned to an integrable function at one point has no effect on its measure. Thus the density limit cannot recover an arbitrarily chosen pointwise representative at every exceptional point. A singular measure also needs separate treatment: an atom at the center contributes a fixed positive numerator as radius shrinks, so its positive-dimensional density ratio diverges. Later, densities of rectifiable geometric measures use the object's own dimension, rather than ambient volume.

A density supported on a closed interval may jump at an endpoint when extended by zero outside. A centered small interval then samples both occupied and empty sides, so its average can approach half the interior limiting value. This does not contradict differentiation, since finitely many endpoints form a null set. The same half-space sampling will reappear in finite-perimeter boundaries, where density one half has a geometric interpretation.

For an m-dimensional geometric measure, we instead study ball mass divided by $\omega_mr^m$. A line in the plane has a finite one-dimensional density at its regular points, while dividing its mass by planar area makes the ratio diverge. The choice is not a cosmetic normalization: it asks a different dimensional question. Covering and differentiation tools help identify which points behave regularly, but they do not infer dimension solely from an ambient-coordinate picture.